i9o6] ROYCE— PRINCIPLES OF THEORETICAL SCIENCE. 99 



VI. 



In an address which I was privile^^ed to make before the St. 

 Louis Congress I pointed out a contribution to this problem which 

 had been suggested and in part carried out by Mr. A. B. Kempe. 

 I have since further pursued the research which Mr. Kempe has 

 initiated, and have published my results in a paper entitled " The 

 Relation of the Principles of Logic to the Foinidations of Geometry," 

 printed in the Transactions of the American Mathematical Society. 

 This is no place to discuss the issues involved in that paper. I 

 want simply to indicate in a very general way one point re- 

 garding the kind of result which seems to me to be already in sight, 

 although the matter is still very incompletely worked out. The 

 different characteristic forms of thought to which I have referred, 

 are distinguished by the various types of relations which these vari- 

 ous forms exemplify. Thus the characteristic ordinal relation of 

 descriptive geometry is the relation called " between " ; and Dr. Veb- 

 len has shown how in terms of this single relation, and of the as- 

 sumption of the existence of appropriate objects or entities, one could 

 state all the principles that are needed as the foundation for geome- 

 try. The characteristic relation of the world of quantity, the re- 

 lation of " greater and less," is a relation which in combination 

 with the triadic relation that is involved in the ordinary operation 

 of addition, is sufficient to give form to the principles of algebraic 

 analysis. In brief, then, each theoretical science has its own char- 

 acteristic set of relationships. When so viewed these relations stand 

 by themselves, as if they were separate facts in the natural history 

 of the forms of thought. Relations may be classified, just as truly 

 as birds, or as bacteria may be classified. There are relations dyadic, 

 triadic, n-adic, there are relations symmetrical, unsymmetrical 

 transitive, intransitive. The^e varieties of form in the world of 

 relation, when thus viewed, seem ultimate and irreducible. Yet 

 I do not think that anybody finds it self-evident, or axiomatic, that 

 only these relations should be possible. I do not think that we have 

 any warrant for saying on the other hand that the sorts of relations 

 which exist are capable of a simply limitless and a capricious variety. 

 The concept of a relation is to my mind, as to the minds of a good 

 many of my colleagues, something that is intelligible only in terms 



